# Bott periodicity theorem

A fundamental theorem in $ K $-
theory which, in its simplest form, states that for any (compact) space $ X $
there exists an isomorphism between the rings $ K(X) \otimes K(S ^ {2} ) $
and $ K(X \times S ^ {2} ) $.
More generally, if $ L $
is a complex vector bundle over $ X $
and $ P(L \oplus 1) $
is the projectivization of $ L \oplus 1 $,
then the ring $ K(P(L \oplus 1)) $
is a $ K(X) $-
algebra with one generator $ [H] $
and a unique relation $ ([H] - [1])([L][H] - [1]) = 0 $,
where $ [E] $
is the image of a vector bundle $ E $
in $ K(X) $
and $ H ^ {-1} $
is the Hopf fibration over $ P(L \oplus 1) $.
This fact is equivalent to the existence of a Thom isomorphism in $ K $-
theory for complex vector bundles. In particular, $ P(1 \oplus 1) = X \times S ^ {2} $.
Bott's periodicity theorem was first demonstrated by R. Bott [1] using Morse theory, and was then re-formulated in terms of $ K $-
theory [6]; an analogous theorem has also been demonstrated for real fibre bundles.

Bott's periodicity theorem establishes the property of the stable homotopy type of the unitary group $ U $, consisting in the fact that $ {\Omega ^ {2} } U \sim U $, where $ \Omega X $ is the space of loops on $ X $, and $ \sim $ is weak homotopy equivalence, in particular $ \pi _ {i} (U) = \pi _ {i+2} (U) $ for $ i = 0, 1 \dots $ where $ \pi _ {i} $ is the $ i $- th homotopy group. Similarly, for the orthogonal group $ O $:

$$ \Omega ^ {8} O \sim O,\ \ \pi _ {i} (O) = \pi _ {i+ 8 } (O). $$

#### References

[1] | R. Bott, "The stable homotopy of the classical groups" Ann. of Math. (2) , 70 : 2 (1959) pp. 313–337 MR0110104 Zbl 0129.15601 |

[2] | J.W. Milnor, "Morse theory" , Princeton Univ. Press (1963) MR0163331 Zbl 0108.10401 |

[3] | M.F. Atiyah, "$K$-theory: lectures" , Benjamin (1967) MR224083 |

[4] | D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) MR0229247 Zbl 0144.44804 |

[5] | J.C. Moore, "On the periodicity theorem for complex vector bundles" , Sem. H. Cartan (1959–1960) |

[6] | M.F. Atiyah, R. Bott, "On the periodicity theorem for complex vector bundles" Acta Math. , 112 (1964) pp. 229–247 MR0178470 Zbl 0131.38201 |

#### Comments

#### References

[a1] | R. Bott, "Lectures on $K(X)$" , Benjamin (1969) MR0258020 Zbl 0194.23904 |

[a2] | M. Karoubi, "$K$-theory" , Springer (1978) MR0488029 Zbl 0382.55002 |

**How to Cite This Entry:**

Bott periodicity theorem.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Bott_periodicity_theorem&oldid=51063